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Properties

Label 1-304-304.75-r0-0-0
Degree $1$
Conductor $304$
Sign $0.923 - 0.382i$
Analytic cond. $1.41177$
Root an. cond. $1.41177$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + i·5-s + 7-s − 9-s − i·11-s + i·13-s + 15-s + 17-s − i·21-s + 23-s − 25-s + i·27-s + i·29-s + 31-s − 33-s + ⋯
L(s)  = 1  − i·3-s + i·5-s + 7-s − 9-s − i·11-s + i·13-s + 15-s + 17-s − i·21-s + 23-s − 25-s + i·27-s + i·29-s + 31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.359853182 - 0.2704916157i\)
\(L(\frac12)\) \(\approx\) \(1.359853182 - 0.2704916157i\)
\(L(1)\) \(\approx\) \(1.161086043 - 0.1777482407i\)
\(L(1)\) \(\approx\) \(1.161086043 - 0.1777482407i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - iT \)
11 \( 1 \)
13 \( 1 + iT \)
17 \( 1 \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 + T \)
59 \( 1 \)
61 \( 1 + T \)
67 \( 1 \)
71 \( 1 \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 \)
89 \( 1 + T \)
97 \( 1 \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.22160837232816197524469003402, −24.724366140499859299772302102687, −23.37858156229460871890183664668, −22.8167372259277174959657686931, −21.525892290104600716436377741793, −20.727847707422366013433712308384, −20.46963780302966202868170791512, −19.29031687279953959891322331551, −17.67915524835358111513931811595, −17.27973318760787905585518807667, −16.263256739729628090935098659, −15.24877941560065584079773426019, −14.69149845118788140381299020787, −13.4246952915863321916112907733, −12.29299804639669970997023516945, −11.4498760776060444458118550722, −10.27222731149147095526099492772, −9.52954022743210192876169260669, −8.39880522856335437945289004809, −7.71142278734656502493862544207, −5.80834362815293301655045351329, −4.91633015671277450739792803141, −4.32536402587824058697723485464, −2.83418476438988663423970719975, −1.21445725035955422966561892690, 1.24760547448037303167887664810, 2.44124780893700280230999549276, 3.54641439575997945880511230167, 5.22327343281503634038598777572, 6.2906644703246743299406314274, 7.20429195646113919664084899098, 8.048472036130073938947391190122, 9.0665800911744695912290379639, 10.6817571137657847007805447494, 11.35494817401835393463081330972, 12.13696435419498614923206340037, 13.51014097974305997149433767212, 14.25806142983636120657651275176, 14.764005945780927589872516062, 16.31689782297062654948082165461, 17.31636325994519474192824481268, 18.18680435967227172783669256487, 18.91302822427969477202159862370, 19.45722025028766078510966136054, 20.94587728733092051800756871705, 21.59984986963251915567905621187, 22.80232035217554442592811306359, 23.57834465532813997205719591174, 24.29397010146231140803826383706, 25.16340637085992836389730154583

Graph of the $Z$-function along the critical line